What is the integral of (((sqrt(ln(x))))/x) ?

The integral of (((sqrt(ln(x))))/x) is 2/3 ln^{3/2}(x)+C

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Find the Derivative of the Integral integral of ( square root of natural log of x)/x with respect to x

\int \frac{( ln ( x ) )}{x} d x

\frac{2}{3} ln^{\frac{3}{2}} (x) + C

Solution steps

\int \frac{ln ( x )}{x} d x

Apply u-substitution

: \int u d u

= \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = ln (x)

= \frac{2}{3} ln^{\frac{3}{2}} (x)

Add a constant to the solution

= \frac{2}{3} ln^{\frac{3}{2}} (x) + C

x \to 1 - lim (\frac{( x - 1 )}{( x ^{2} - 4 )})

(- 3, 3), (5, 9)

x \to - \infty lim (\frac{( x ^{4} - 1 )}{( x ^{3} - 1 )})

\int (\frac{( x ^{2} - 9 )}{( x ^{3} )}) d x

\int (4 - x^{2}) d x

What is the integral of (((sqrt(ln(x))))/x) ?
The integral of (((sqrt(ln(x))))/x) is 2/3 ln^{3/2}(x)+C